Designing an efficient facial recognition system is a challenging task that has received renewed attention in recent years due to its wide applicability in security, control and biomedical applications. The quality of a recognition system depends on several factors such as the computational complexity, the recognition rate and the storage requirements.
A large number of methods have been proposed for facial recognition. Facial recognition based on Two-Dimensional Independent Component Analysis (2D ICA) and Fuzzy Supported Vector Machine (FSVM) was proposed in Y. Liu et al. Y. Liu et al., “Face Recognition Based on Independent Component Analysis and Fuzzy Support Vector Machine,” IEEE 6th World Congress on Intelligent Control and Automation, Dalian China, Vol. 2, pp 9889-9892, June 2006. There, the extracted features were obtained by applying 2D ICA to the low-frequency band of different levels of Two-Dimensional Discrete Wavelet Transform (2D DWT) based on Daubechies 4. Then, the extracted features were fed to FSVM for classification. The method was tested on the ORL and YALE databases.
In M. Li et al., 2D DWT, 2D ICA and Radial Basis Function (RBF) were used for facial recognition. M. Li et al., “Face Recognition Based on WT, FastICA and RBF Neural Network,” IEEE 3rd ICNC, Haikou China, Vol. 02, pp. 3-7, August 2007. The 2D DWT based on Daubechies 4 was used to extract the useful information from the face images and then 2D ICA was used to get discriminating and independent features. Then, the extracted features were fed to a RBF for the recognition task. The method was tested on the ORL database and produced a recognition rate of 93%.
Three different methods based on Wavelet, Fourier, and Discrete Cosine (DCT) transforms for facial recognition were studied in Z. Lihong et al. “Face Recognition Based on Image Transformation,” IEEE GCIS, Xiamen China, Vol. 04, pp. 418-421, May 2009. Each transform was applied individually to find the features. Nearest-neighbor classifiers using two similarity criteria, Euclidean distance and correlation coefficients, were used for the recognition task. Each transform was tested on the ORL and YALE databases and the highest recognition rates achieved were 97% and 94.5% for the ORL and YALE databases, respectively.
Multiresolution analysis based on Independent Component Analysis (ICA) was proposed in K. S. Kinage et al. K. S. Kinage et al., “Face Recognition Based on Independent Component Analysis on Wavelet Subband,” IEEE 3rd ICCSIT, Chengdu China, Vol. 9, pp. 436-440, July 2010. There, various wavelet functions were applied to extract the features from the images by decomposing the face image into eight levels. Then, the extracted facial features were analyzed using 2D ICA and the Euclidean distance was used to measure accuracy. The method was tested on the ORL database and the recognition rate recorded was 91.5%.
In Y. Hongtao et al., DCT and SVM were used for facial recognition. Y. Hongtao et al., “Face Recognition with Discrete Cosine Transform,” IEEE 2nd IMCCC, Harbin China, pp. 802-805, December 2012. The feature coefficients were extracted using DCT and then fed to SVM for recognition. The system was tested on the ORL database and produced a recognition rate of 95%.
Discrete Multiwavelet Transform (DWT) is broadly applied in computer vision, image processing, signal processing, and pattern recognition. In contrast to DWT, which uses one scaling function Φ(t) and one wavelet function Ψ(t), Two-Dimensional Discrete Multiwavelet Transform (2D DMWT) is based on Multiresolution Analysis (MRA) and uses multiple scaling and wavelet functions. The scaling and wavelet functions are associated with low and high pass filters, respectively. In multiwavelets, the N scaling and wavelet functions can be expressed in vector form as:Φ(t)≡[ϕ1(t),ϕ2(t), . . . ,ϕN(t)]Ψ(t)≡[ψ1(t),ψ2(t), . . . ,ψN(t)]where Φ(t) and Ψ(t) are multi-scaling and wavelet functions, respectively. P. V. N. Reddy et al., “Multiwavelet Based Texture Features for Content Based Image Retrieval,” International Journal of Computer Science and Technology, Vol. 2, Issue 1, March 2011, pp. 141-145. Note that N=1 corresponds to the standard wavelet transform. Akin to the scalar wavelet, it is possible to write the two-scale dilation and wavelet equations as:
            Φ      ⁡              (        t        )              =                  2            ⁢                        ∑                      k            =                          -              ∞                                ∞                ⁢                                  ⁢                              H            k                    ·                      Φ            ⁡                          (                                                2                  ⁢                                                                          ⁢                  t                                -                k                            )                                                              Ψ        ⁡                  (          t          )                    =                        2                ⁢                              ∑                          k              =                              -                ∞                                      ∞                    ⁢                                          ⁢                                    G              k                        ·                          Φ              ⁡                              (                                                      2                    ⁢                                                                                  ⁢                    t                                    -                  k                                )                                                          ,  where Hk and Gk are the N×N filter matrices for each integer k. In contrast to DWT, multiwavelets have several favorable features such as orthogonality, symmetry, compact support, and high order of approximation. These features provide more degrees of freedom compared to the wavelet transform, which cannot possess all these features at the same time.
Multiwavelets typically use a multiplicity N=2, in which case the scaling and wavelet functions can be written as Φ(t)=[ϕ1(t) ϕ2(t)]T and Ψ(t)=[ψ1(t) ψ2(t)]T, respectively. Here, the Gernimo, Hardin and Massopust (GHM) filter introduced by Gernimo et al. is used. Gernimo et al. “Fractal Functions and Wavelet Expansions Based on several Scaling Functions,” Journal of Approximation Theory, vol. 78, September 1994, pp. 373-401. For the GHM case, the multi-scaling function Hk has four scaling matrices H0, H1, H2, and H3 and also Gk has four wavelet matrices G0, G1, G2 and G3 as shown below:
                    H        0            =              [                                                            3                                  5                  ⁢                                      2                                                                                                      2                5                                                                                                          -                  1                                20                                                                                      -                  2                                                  10                  ⁢                                      2                                                                                      ]              ,                  H        1            =              [                                                            3                                  2                  ⁢                                      2                                                                                      0                                                                          9                20                                                                    1                                  2                                                                    ]                                H        2            =              [                                            0                                      0                                                                          9                20                                                                                      -                  3                                                  10                  ⁢                                      2                                                                                      ]              ,                  H        3            =              [                                            0                                      0                                                                                            -                  1                                20                                                    0                                      ]                                G        0            =              [                                                                              -                  1                                20                                                                                      -                  3                                                  10                  ⁢                                      2                                                                                                                          1                                  10                  ⁢                                      2                                                                                                      3                10                                                    ]              ,                  G        1            =              [                                                            9                20                                                                                      -                  1                                                  2                                                                                                        9                20                                                                    1                                  2                                                                    ]                                G        2            =              [                                                            9                20                                                                                      -                  3                                                  10                  ⁢                                      2                                                                                                                          9                                  10                  ⁢                                      2                                                                                                                        -                  3                                10                                                    ]              ,                  G        3            =              [                                                                              -                  1                                20                                                    0                                                                                            -                  1                                                  10                  ⁢                                      2                                                                                      0                                      ]            V. Strela and A Walden, “Orthogonal and Biorthogonal Multiwavelets for Signal Denoising and Image Compression,” Proceedings of SPIE, Issue 1, November 1998, pp 96-107.
These matrices are used to construct the transformation matrix T which is used in the 2D DMWT to extract the useful information from the face images. The transformation matrix T can be written as
  T  =      [                                        H            0                                                H            1                                                H            2                                                H            3                                    0                          0                          …                                                  G            0                                                G            1                                                G            2                                                G            3                                    0                          0                          …                                      0                          0                                      H            0                                                H            1                                                H            2                                                H            3                                    …                                      0                          0                                      G            0                                                G            1                                                G            2                                                G            3                                    …                                      ⋮                          ⋮                          ⋮                          ⋮                          ⋮                          ⋮                          …                      ]  Further details about 2D DMWT can be found in V. Strela and A. Walden and V. Strela et al. V. Strela et al., “The Application of Multiwavelet Filterbanks to Image Processing,” IEEE Tran. On Image Processing, Vol. 8, No. 4, April 1999, pp. 548-563.
ICA is widely used to deal with problems similar to the cocktail-party problem. A. J. Bell et al., “An Information-Maximization approach to blind separation and blind deconvolution,” Elsevier Journal of Neural Computation, Vol. 7, Issue 6, November 1995, pp. 1129-1159. Hence, it is extensively used for Blind Signal Separation (BSS). A. J. Bell et al.; A. Hyvarinen, “Fast and Robust Fixed-Point Algorithms for Independent Component Analysis,” IEEE Tran. on Neural Network, Vol. 10, Issue 3, May 1999, pp. 626-634. However, one of the most attractive applications is in feature extraction, wherein ICA can be used to extract independent image bases that are not necessarily orthogonal. M. Li et al.
ICA is a statistical signal processing technique that is very sensitive to high order statistics. It can be considered as a generalization of Principal Component Analysis (PCA). The idea of ICA is to represent a set of random variables using basis functions such that the components are statistically independent or nearly independent. Consider observing M random variables X1, X2, . . . XM with zero mean, which are assumed to be linear combinations of N mutually independent components S1, S2, . . . SN. Let the vector X=[X1, X2, . . . XM]T denote a M×1 vector of observed variables and S=[s1, s2, . . . sN]T an N×1 vector of mutually independent components. The relation between X and S is expressed as:X=A·S  (5)where A is a full rank M×N unknown matrix, called the mixing matrix or the feature matrix. P. C. Yuen, et al., “Face Representation Using Independent Component Analysis,” Elsevier Journal of The Pattern Recognition Society, Vol. 35, Issue 6, June 2002, pp. 1247-1257. In feature extraction, the columns of A are the features and the component si is the coefficient of the ith feature in the data vector X.
Nevertheless, ICA has two main drawbacks. First, it typically requires complex matrix operations. P. Comon, “Independent Component Analysis, A new Concept?,” Elsevier Journal of Signal Processing, Vol. 36, Issue 3, April 1994, pp. 287-314. Second, it has slow convergence. A. J. Bell et al. Therefore, A. Hyvarinen introduced a new method called FastICA. FastICA is computationally more efficient for estimating the ICA components and has a faster convergence rate by using a fixed point iteration method. Further detail about FastICA can be found in A. Hyvarinen and M. S. Bartlett et al. M. S. Bartlett, et al., “Face Recognition by Independent Component Analysis,” IEEE Tran. on Neural Networks, Vol. 13, Issue 6, November 2002, pp. 1450-1464.